In [1]}, Białynicki-Birula proved that, on a smooth scheme \(X\)  with \(\mathbb {C}^\ast \)  action, the attracting subset of \(X\)  (i.e. the subset of points \(x\in X\)  such that \(\lim _{t\rightarrow 0}t.x\)  exists) admits a sort of cellular decomposition. Each cell of this so-called Białynicki-Birula decomposition is an affine fiber bundle on a component of the fixed variety \(X^0\) , whose dimension is given by the number of contracting \(\mathbb {C}^\ast \) -weights in the tangent space of \(X^0\) . Hence one can compute the cohomology of the attracting subset of \(X\)  in terms of the cohomology of \(X^0\) . In [2]} and [3]}, Braden and Drinfeld gave a functorial reformulation of the Białynicki-Birula decomposition, which can be extended to singular algebraic spaces or stacks. For any algebraic space (resp. stack) \(X\)  with \(\mathbb {C}^\ast \) -action one can define the \(\mathbb {C}^\ast \) -fixed space (resp. stack) \(X^0\)  and also the contracting (resp. repelling) space (resp. stack) \(X^\pm \)  using the functor of points in an obvious way. There is then the so-called hyperbolic localization diagram:
\(\begin{tikzcd}X & X^\pm [l,swap,"\eta ^\pm _X"][r,"p^\pm _X"] & X^0\end{tikzcd}\) 
